Limits and Derivatives

Limits and Derivatives

Question

Find the derivative of the following :


space space space space space space space space fraction numerator 2 over denominator straight x plus 1 end fraction minus fraction numerator straight x squared over denominator 3 straight x minus 1 end fraction

Answer

Let               space space space space space space space space space space space space space straight y space equals space fraction numerator 2 over denominator straight x plus 1 end fraction minus fraction numerator straight x squared over denominator 3 straight x minus 1 end fraction
Differentiating both sides w.r.t. x, we get

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   space space space space space space space space space space space space space space equals space 2 space straight d over dx left parenthesis straight x plus 1 right parenthesis to the power of negative 1 end exponent minus fraction numerator left parenthesis 3 straight x minus 1 right parenthesis begin display style straight d over dx end style straight x squared minus straight x squared space begin display style straight d over dx end style left parenthesis 3 straight x minus 1 right parenthesis over denominator left parenthesis 3 straight x minus 1 right parenthesis squared end fraction
     space space space space space space space space space space space space space space space space space space equals space fraction numerator negative 2 over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction minus fraction numerator 6 straight x squared minus 2 straight x minus 3 straight x squared over denominator left parenthesis 3 straight x minus 1 right parenthesis end fraction equals fraction numerator negative 2 over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction minus fraction numerator 3 straight x squared minus 2 straight x over denominator left parenthesis 3 straight x minus 1 right parenthesis end fraction
  

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